I don't know what you mean by "polynomial series" since your function doesn't seem to have much to do with polynomials (perhaps you could elaborate?).

$S(x) = -\frac12 \theta(\frac12, \frac{\log x}{\pi i }) - 1$, where $\theta$ is [Jacobi's theta function][1].  I don't know of any nice algebraic methods to take inverses of modular forms (even if you restrict to real $x$, i.e., $\tau$ purely imaginary), so I'm not sure if this is the sort of answer you want.


  [1]: http://en.wikipedia.org/wiki/Theta_function